(Almost) quadruply optimal unitary designs in 1D

We construct n-qubit approximate unitary k-designs in 1D systems, achieving circuit depth O\(log(n/varepsilon\) + klog k) with relative error varepsilon and requiring O\(nklog k\) magic gates. This matches existing lower bounds Ω\(log(n/varepsilon\) + k) for circuit depth, and widetildeΩ(nk) for the required number of T gates, up to a log k factor, achieving simultaneous near-optimality in all parameters. Our construction is based on a combination and refinement of two existing results. We reduce the required magic block size for breaking Clifford symmetries in the magic-augmented circuit construction of Zhang et al. from O\(klog k\) to O\(log k\). We also improve the breakthrough construction of Chen et al. to construct a generating set of 1D local constant-depth circuits for the unitary group with a constant spectral gap, making O\(log k\)-local random unitaries realizable in depth O\(klog k\). As a by-product, we provide a constant-size 1D-local generating set for the Clifford group, which we expect to be of independent interest. Combining the two results with the gluing lemma, we prove the final result.